We wish to constrain the cosmic-ray proton (CRp) population in
galaxy clusters. By hadronic interactions with the thermal gas of the
intra-cluster medium (ICM), the CRp produce γ-rays for which we
develop an analytic formalism to deduce their spectral distribution.
Assuming the CRp-to-thermal energy density ratio and the CRp
spectral index to be spatially constant, we derive an analytic relation
between the γ-ray and bolometric X-ray fluxes,
and . Based on our relation, we compile a sample of suitable
clusters which are promising candidates for future detection of γ-rays
resulting from hadronic CRp interactions. Comparing to EGRET upper limits, we
constrain the CRp population in the cooling flow clusters Perseus and Virgo
to . Assuming a plausible value for the CRp diffusion
coefficient κ, we find the central CRp injection luminosity of M 87 to
be limited to . The synchrotron emission from secondary electrons
generated in CRp hadronic interactions allows even tighter limits to be
placed on the CRp population using radio observations. We obtain excellent
agreement between the observed and theoretical radio brightness profiles for
Perseus, but not for Coma without a radially increasing CRp-to-thermal energy
density profile. Since the CRp and magnetic energy densities necessary to
reproduce the observed radio flux are very plausible, we propose synchrotron
emission from secondary electrons as an attractive explanation of the radio
mini-halos found in cooling flow clusters. This model can be tested with
future sensitive γ-ray observations of the accompanying
-decays. We identify Perseus (A 426), Virgo, Ophiuchus, and Coma
(A 1656) as the most promising candidate clusters for such observations.

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